Upsetting the Foundations for Mathematics

Foundations of Mathematics for the Working Mathematician

Foundations of Mathematics

The aim of this book is to provide a thorough orientation to the current content of Hilbert’s proof theory. Despite the fact that the achievements to date have been modest in comparison to the goals of the theory, there are still plenty of suggestive results, viewpoints, and proof ideas that seem worth reporting on. The purpose of the book has resulted in two main themes for the contents of thi...

Formalising foundations of mathematics

Over the recent decades there has been a trend towards formalized mathematics, and a number of sophisticated systems have been developed to support the formalization process and mechanically verify its result. However, each tool is based on a speci c foundation of mathematics, and formalizations in di erent systems are not necessarily compatible. Therefore, the integration of these foundations ...

Evolutionary Foundations of Mathematics

We propose a simple cognitive model where qualitative and quantitative comparisons enable animals to identify objects, associate them with their properties held in memory and make naive inference. Simple notions like equivalence relations, order relations are used. We then show that such processes are at the root of human mathematical reasoning by showing that the elements of totally ordered se...

Foundations of mathematics

A tuple (or n-tuple, for any integer n) is an interpretation of a list of n variables. It is thus a meta-function from a finite meta-set, to the universe. Tuples of a given kind (list of variables with their types) can be added to any theory as a new type of objects, whose variables are meant as abbreviations of packs of n variables (copies of the list with the same old types) x = (x0, · · · , ...